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Full-Text Articles in Mathematics
Zadeh's Vision Of Going From Fuzzy To Computing With Words: From The Idea's Origin To Current Successes To Remaining Challenges, Vladik Kreinovich
Zadeh's Vision Of Going From Fuzzy To Computing With Words: From The Idea's Origin To Current Successes To Remaining Challenges, Vladik Kreinovich
Departmental Technical Reports (CS)
No abstract provided.
Why Clayton And Gumbel Copulas: A Symmetry-Based Explanation, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta
Why Clayton And Gumbel Copulas: A Symmetry-Based Explanation, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta
Departmental Technical Reports (CS)
In econometrics, many distributions are non-Gaussian. To describe dependence between non-Gaussian variables, it is usually not sufficient to provide their correlation: it is desirable to also know the corresponding copula. There are many different families of copulas; which family shall we use? In many econometric applications, two families of copulas have been most efficient: the Clayton and the Gumbel copulas. In this paper, we provide a theoretical explanation for this empirical efficiency, by showing that these copulas naturally follow from reasonable symmetry assumptions. This symmetry justification also allows us to provide recommendations about which families of copulas we should use …
If Energy Is Not Preserved, Then Planck's Constant Is No Longer A Constant: A Theorem, Vladik Kreinovich, Andres Ortiz
If Energy Is Not Preserved, Then Planck's Constant Is No Longer A Constant: A Theorem, Vladik Kreinovich, Andres Ortiz
Departmental Technical Reports (CS)
For any physical theory, to experimentally check its validity, we need to formulate an alternative theory and check whether the experimental results are consistent with the original theory or with an alternative theory. In particular, to check whether energy is preserved, it is necessary to formulate an alternative theory in which energy is not preserved. Formulating such a theory is not an easy task in quantum physics, where the usual Schroedinger equation implicitly assumes the existence of an energy (Hamiltonian) operator whose value is preserved. In this paper, we show that the only way to get a consistent quantum theory …
Towards Unique Physically Meaningful Definitions Of Random And Typical Objects, Luc Longpre, Olga Kosheleva
Towards Unique Physically Meaningful Definitions Of Random And Typical Objects, Luc Longpre, Olga Kosheleva
Departmental Technical Reports (CS)
To distinguish between random and non-random sequence, Kolmogorov and Martin-Lof proposed a new definition of randomness, according to which an object (e.g., a sequence of 0s and 1s) if random if it satisfies all probability laws, i.e., in more precise terms, if it does not belong to any definable set of probability measure 0. This definition reflect the usual physicists' idea that events with probability 0 cannot happen. Physicists -- especially in statistical physics -- often claim a stronger statement: that events with a very small probability cannot happen either. A modification of Kolmogorov-Martin-Lof's (KLM) definition has been proposed to …
In Applications, A Rigorous Proof Is Not Enough: It Is Also Important To Have An Intuitive Understanding, Vladik Kreinovich
In Applications, A Rigorous Proof Is Not Enough: It Is Also Important To Have An Intuitive Understanding, Vladik Kreinovich
Departmental Technical Reports (CS)
From a purely mathematical viewpoint, once a statement is rigorously proven, it should be accepted as true. Surprisingly, in applications, users are often reluctant to accept a rigorously proven statement until the proof is supplemented by its intuitive explanation. In this paper, we show that this seemingly unreasonable reluctance makes perfect sense: the proven statement is about the mathematical model which is an approximation to the actual system; an intuitive explanation provides some confidence that the statement holds not only for the model, but also for systems approximately equal to this model -- in particular, for the actual system of …
Kansei Engineering: Towards Optimal Set Of Designs, Van-Nam Huynh, Octavio Lerma, Vladik Kreinovich
Kansei Engineering: Towards Optimal Set Of Designs, Van-Nam Huynh, Octavio Lerma, Vladik Kreinovich
Departmental Technical Reports (CS)
In many engineering situations, we need to take into account subjective user preferences; taking such preference into account is known as {\em Kansei Engineering}. In this paper, we formulate the problem of selecting optimal set of designs in Kansei engineering as a mathematical optimization problem, and we provide an explicit solution to this optimization problem.
Possible And Necessary Orders, Equivalences, Etc.: From Modal Logic To Modal Mathematics, Francisco Zapata, Olga Kosheleva
Possible And Necessary Orders, Equivalences, Etc.: From Modal Logic To Modal Mathematics, Francisco Zapata, Olga Kosheleva
Departmental Technical Reports (CS)
In practice, we are often interested in order relations (e.g., when we describe preferences) or equivalence relations (e.g., when we describe clustering). Often, we do not have a complete information about the corresponding relation; as a result, we have several relations consistent with our knowledge. In such situations, it is desirable to know which elements a and b are possibly connected by the relation and which are necessarily connected by this relation. In this paper, we provide a full description of all such possible and necessary orders and equivalence relations. For example, possible orders are exactly reflexive relations, while necessary …
Orders On Intervals Over Partially Ordered Sets: Extending Allen's Algebra And Interval Graph Results, Francisco Zapata, Vladik Kreinovich, Cliff Joslyn, Emilie Hogan
Orders On Intervals Over Partially Ordered Sets: Extending Allen's Algebra And Interval Graph Results, Francisco Zapata, Vladik Kreinovich, Cliff Joslyn, Emilie Hogan
Departmental Technical Reports (CS)
To make a decision, we need to compare the values of quantities. In many practical situations, we know the values with interval uncertainty. In such situations, we need to compare intervals. Allen's algebra describes all possible relations between intervals on the real line which are generated by the ordering of endpoints; ordering relations between such intervals have also been well studied. In this paper, we extend this description to intervals in an arbitrary partially ordered set (poset). In particular, we explicitly describe ordering relations between intervals that generalize relation between points. As auxiliary results, we provide a logical interpretation of …
How To Define Mean, Variance, Etc., For Heavy-Tailed Distributions: A Fractal-Motivated Approach, Vladik Kreinovich, Olga Kosheleva
How To Define Mean, Variance, Etc., For Heavy-Tailed Distributions: A Fractal-Motivated Approach, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
In many practical situations, we encounter heavy-tailed distributions for which the variance -- and even sometimes the mean -- are infinite. We propose a fractal-motivated approach that enables us to gauge the mean and variance of such distributions.
From Unbiased Numerical Estimates To Unbiased Interval Estimates, Baokun Li, Gang Xiang, Vladik Kreinovich, Panagios Moscopoulos
From Unbiased Numerical Estimates To Unbiased Interval Estimates, Baokun Li, Gang Xiang, Vladik Kreinovich, Panagios Moscopoulos
Departmental Technical Reports (CS)
One of the main objectives of statistics is to estimate the parameters of a probability distribution based on a sample taken from this distribution. Of course, since the sample is finite, the estimate X is, in general, different from the actual value x of the corresponding parameter. What we can require is that the corresponding estimate is unbiased, i.e., that the mean value of the difference X - x is equal to 0: E[X] = x. In some problems, unbiased estimates are not possible. We show that in some such problems, it is possible to have interval unbiased estimates, i.e., …
Decision Making Under Interval And Fuzzy Uncertainty: Towards An Operational Approach, Rafik Aliev, Oleg H. Huseynov, Vladik Kreinovich
Decision Making Under Interval And Fuzzy Uncertainty: Towards An Operational Approach, Rafik Aliev, Oleg H. Huseynov, Vladik Kreinovich
Departmental Technical Reports (CS)
Traditional decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them is preferable. In reality, often, when the alternatives are close, the user is either completely unable to select one of these alternatives, or selects one of the alternatives only "to some extent". How can we extend the traditional decision theory to such realistic interval and fuzzy cases? In their previous papers, the first two authors proposed a natural generalization of the usual decision theory axioms to interval and fuzzy cases, and described decision coming from this generalization. …
Membership Functions Or Alpha-Cuts? Algorithmic (Constructivist) Analysis Justifies An Interval Approach, Vladik Kreinovich
Membership Functions Or Alpha-Cuts? Algorithmic (Constructivist) Analysis Justifies An Interval Approach, Vladik Kreinovich
Departmental Technical Reports (CS)
In his pioneering papers, Igor Zaslavsky started an algorithmic (constructivist) analysis of fuzzy logic. In this paper, we extend this analysis to fuzzy mathematics and fuzzy data processing. Specifically, we show that the two mathematically equivalent representations of a fuzzy number -- by a membership function and by alpha-cuts -- are not algorithmically equivalent, and only the alpha-cut representation enables us to efficiently process fuzzy data.
Estimating Correlation Under Interval And Fuzzy Uncertainty: Case Of Hierarchical Estimation, Ali Jalal-Kamali
Estimating Correlation Under Interval And Fuzzy Uncertainty: Case Of Hierarchical Estimation, Ali Jalal-Kamali
Departmental Technical Reports (CS)
In many situations, we are interested in finding the correlation ρ between different quantities x and y based on the values xi and yi of these quantities measured in different situations i. The correlation is easy to compute when we know the exact sample values xi and yi. In practice, the sample values come from measurements or from expert estimates; in both cases, the values are not exact. Sometimes, we know the probabilities of different values of measurement errors, but in many cases, we only know the upper bounds Δxi and Δyi on …
How To Define Average Class Size (And Deviations From The Average Class Size) In A Way Which Is Most Adequate For Teaching Effectiveness, Olga Kosheleva, Vladik Kreinovich
How To Define Average Class Size (And Deviations From The Average Class Size) In A Way Which Is Most Adequate For Teaching Effectiveness, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
When students select a university, one of the important parameters is the average class size. This average is usually estimated as an arithmetic average of all the class sizes. However, it has been recently shown that to more adequately describe students' perception of a class size, it makes more sense to average not over classes, but over all students -- which leads to a different characteristics of the average class size. In this paper, we analyze which characteristic is most adequate from the viewpoint of efficient learning. Somewhat surprisingly, it turns out that the arithmetic average is the most adequate …
Semi-Heuristic Target-Based Fuzzy Decision Procedures: Towards A New Interval Justification, Christian Servin, Van-Nam Huynh, Yoshiteru Nakamori
Semi-Heuristic Target-Based Fuzzy Decision Procedures: Towards A New Interval Justification, Christian Servin, Van-Nam Huynh, Yoshiteru Nakamori
Departmental Technical Reports (CS)
To more adequately describe human decision making, V.-N. Nuynh, Y. Nakamori, and others proposed a special semi-heuristic target-based fuzzy decision procedure. A usual justification for this procedure is based on the selection of the simplest possible membership functions and "and"- and "or"-operations; if we use more complex membership functions and "and"- and "or"-operations, we get different results. Interestingly, in practical applications, the procedure based on the simplest choices most adequately describes human preferences. It is therefore desirable to come up with a justification that explains this empirical fact. Such a justification is proposed in this paper
Simplicity Is Worse Than Theft: A Constraint-Based Explanation Of A Seemingly Counter-Intuitive Russian Saying, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich
Simplicity Is Worse Than Theft: A Constraint-Based Explanation Of A Seemingly Counter-Intuitive Russian Saying, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, simplified models, models that enable us to gauge the quality of different decisions reasonably well, lead to far-from-optimal situations when used in searching for an optimal decision. There is even an appropriate Russian saying: simplicity is worse than theft. In this paper, we provide a mathematical explanation of this phenomenon.
Kinematic Spaces And De Vries Algebras: Towards Possible Physical Meaning Of De Vries Algebras, Olga Kosheleva, Francisco Zapata
Kinematic Spaces And De Vries Algebras: Towards Possible Physical Meaning Of De Vries Algebras, Olga Kosheleva, Francisco Zapata
Departmental Technical Reports (CS)
Traditionally, in physics, space-times are described by (pseudo-)Riemann spaces, i.e., by smooth manifolds with a tensor metric field. However, in several physically interesting situations smoothness is violated: near the Big Bang, at the black holes, and on the microlevel, when we take into account quantum effects. In all these situations, what remains is causality -- an ordering relation. To describe such situations, in the 1960s, geometers H. Busemann and R. Pimenov and physicists E. Kronheimer and R. Penrose developed a theory of kinematic spaces. Originally, kinematic spaces were formulated as topological ordered spaces, but it turned out that kinematic …